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JOURNALS // Izvestiya of Saratov University. Mathematics. Mechanics. Informatics // Archive

Izv. Saratov Univ. Math. Mech. Inform., 2022 Volume 22, Issue 4, Pages 416–429 (Mi isu953)

Scientific Part
Mathematics

Representation of functions on a line by a series of exponential monomials

A. S. Krivosheeva, O. A. Krivosheevab

a Institute of Mathematics with Computing Centre, Ufa Federal Research Center, RAS, 112 Chernyshevsky St., Ufa 450008, Russia
b Bashkir State University, 32 Zaki Validi St., Ufa 450076, Russia

Abstract: In this work, we consider the weight spaces of integrable functions $L_p^\omega$ ($p\geq 1$) and continuous functions $C^\omega$ on the real line. Let $\Lambda=\{\lambda_k,n_k\}$ be an unbounded increasing sequence of positive numbers $\lambda_k$ and their multiplicities $n_k$, $\mathcal{E}(\Lambda)=\{t^n e^{\lambda_k t}\}$ be a system of exponential monomials constructed from the sequence $\Lambda$. We study the subspaces $W^p (\Lambda,\omega)$ and $W^0 (\Lambda,\omega)$, which are the closures of the linear span of the system $\mathcal{E}(\Lambda)$ in the spaces $L_p^\omega$ and $C^\omega$, respectively. Under natural constraints on $\Lambda$ (the finiteness of the condensation index $S_\Lambda$ and $n_k/\lambda_k\leq c$, $k\geq 1$) and on the convex weight $\omega$, conditions are obtained under which each function of these subspaces continues to an entire function and is represented by a series in the system $\mathcal{E}(\Lambda)$ that converges absolutely and uniformly on compact sets in the plane. In contrast to the previously known results for the specified representation problem, we do not require that the sequence $\Lambda$ has a density, and we do not impose the separability condition: $\lambda_{k+1}-\lambda_k\geq h$, $k\geq 1$ (instead, the condition of equality to zero of the special condensation index is used).

Key words: series of exponential monomials, weight space, analytic continuation, condensation index.

UDC: 517.98

Received: 18.03.2022
Revised: 15.04.2022

DOI: 10.18500/1816-9791-2022-22-4-416-429



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